Quantum gravity,random geometry and critical phenomena

We discuss the theory of non-critical strings with extrinsic curvature embedded in a target space dimensiond greater than one. We emphasize the analogy between 2d gravity coupled to matter and non self-avoiding liquid-like membranes with bending rigidity. We first outline the exact solution for strings in dimensionsd<1 via the double scaling limit of matrix models and then discuss the difficulties of an extension tod>1. Evidence from recent and ongoing numerical simulations of dynamically triangulated random surfaces indicate that there is a non-trivial crossover from a crumpled to an extended surface as the bending rigidity is increased. If the cross-over is a true second order phase transition corresponding to a critical point there is the exciting possibility of obtaining a well defined continuum string theory ford>1. This essay received the third award from the Gravity Research Foundation, 1992-Ed.